(i) Let Sn = 5 + 55 + 555 + ..... to n terms, which can be written as
Sn =

5

9

[9 + 99 + 999 + ... to n terms]
=

5

9

[(10 - 1) + (100 - 1) + (1000 - 1) + ... to n terms]
=

5

9

[(10 + 100 + 1000 + ... to n terms) - (1 + 1 + ... to n terms)]
=

5

9

[

10(10n−1)

10−1

−n] =

5

9

[

10

9

(10n−1)−n] =

50

81

(10n−1)−

5n

9

(ii) Let Sn = 0.6 + 0.66 + 0.666 + ...... to n terms, which can be written as
Sn =

6

9

[0.9 + 0.99 + 0.999 + ... to n terms]
=

6

9

[(1 - 0.1) + (1 - 0.01) + (1 - 0.001) + ... to n terms]
=

6

9

[(1 + 1 + ... to n terms) - (0.1 + 0.01 + 0.001 + ... to n terms)]
=

2

3

[n−0.1

(1−(0.1)n)

1−0.1

] =

2

3

[n−

1

9

(1−

1

10n

)] =

2n

3

−

2

27

(1−

1

10n

)